MinT - Best Known (13, 13+55, s)-Nets in Base 64

Best Known (13, 13+55, s)-Nets in Base 64

(13, 13+55, 177)-Net over F₆₄ — Constructive and digital

Digital (13, 68, 177)-net over F₆₄, using

t-expansion [i] based on digital (7, 68, 177)-net over F₆₄, using
- net from sequence [i] based on digital (7, 176)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 7 and N(F) ≥ 177, using
    - a function field by GarcÃa and Quoos [i]

(13, 13+55, 192)-Net in Base 64 — Constructive

(13, 68, 192)-net in base 64, using

2 times m-reduction [i] based on (13, 70, 192)-net in base 64, using
- base change [i] based on digital (3, 60, 192)-net over F₁₂₈, using
  - net from sequence [i] based on digital (3, 191)-sequence over F₁₂₈, using
    - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₂₈ with g(F) = 3 and N(F) ≥ 192, using
      - a function field by SÃ©mirat [i]

(13, 13+55, 257)-Net over F₆₄ — Digital

Digital (13, 68, 257)-net over F₆₄, using

t-expansion [i] based on digital (12, 68, 257)-net over F₆₄, using
- net from sequence [i] based on digital (12, 256)-sequence over F₆₄, using
  - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₆₄ with g(F) = 12 and N(F) ≥ 257, using
    - K_1,2 from the tower of function fields by Niederreiter and Xing based on the tower by GarcÃa and Stichtenoth over F₆₄ [i]

(13, 13+55, 5247)-Net in Base 64 — Upper bound on s

There is no (13, 68, 5248)-net in base 64, because

1 times m-reduction [i] would yield (13, 67, 5248)-net in base 64, but
- the generalized Rao bound for nets shows that 64^mÂ â‰¥ 10 351329 441668 361658 450029 254278 680962 303651 184366 522201 796751 757025 885318 298374 986021 464337 334047 020138 401209 653905 204633 > 64⁶⁷ [i]